In classical and quantum spin ice, rodlike features appear in the neutron-scattering structure factor when the pinch points characteristic of classical spin ice get washed out. We show that these features do not indicate the absence of spin correlations between planes perpendicular to the rods. Rather, they arise because neutron scattering is largely insensitive to the three-dimensional correlations which are present throughout. We present two very simple models which exhibit a pristine incarnation of such scattering rods. This provides a physical picture for their appearance, elucidates the role played by monopole excitations, and identifies conditions conducive to their observation.